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A Solvable Four-Dimensional QFT

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Quantum Mathematical Physics

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Abstract

We review a sequence of papers in which we show that the quartic matrix model with an external matrix is exactly solvable in terms of the solution of a non-linear integral equation. The interacting scalar model on four-dimensional Moyal space is of this type, and our solution leads to the construction of Schwinger functions. Taking a special limit leads to a QFT on \(\mathbb{R}^{4}\) which satisfies growth property, covariance and symmetry. There is numerical evidence for reflection positivity of the 2-point function for a certain range of the coupling constant.

Mathematics Subject Classification (2010). 81T16, 81T08, 39A99, 45E05

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Notes

  1. 1.

    For each assignment \(N\mapsto f_{N} \in \mathcal{S}^{N}\) of test functions, one has

    $$\displaystyle{\sum _{M,N}\int \!dx\,dy\;S(x_{1},\ldots,x_{N},y_{1},\ldots,y_{M})\overline{f_{N}(x_{1}^{r},\ldots,x_{N}^{r})}f_{M}(y_{1},\ldots,y_{M}) \geq 0\;,}$$

    where \((x^{0},x^{1},\ldots x^{D-1})^{r}:= (-x^{0},x^{1},\ldots x^{D-1})\).

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Acknowledgements

We would like to cordially thank Jürgen Tolksdorf and Felix Finster for invitation to Quantum Mathematical Physics and hospitality during the conference in Regensburg. We submit our best wishes to Eberhard Zeidler.

Author information

Authors and Affiliations

  1. Fakultät für Physik, Universität Wien, Boltzmanngasse 5, A-1090, Wien, Austria

    Harald Grosse

  2. Mathematisches Institut der Westfälischen Wilhelms-Universität, Einsteinstraße 62, D-48149, Münster, Germany

    Raimar Wulkenhaar

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  1. Harald Grosse
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Correspondence to Harald Grosse .

Editor information

Editors and Affiliations

  1. Universität Regensburg, Fakultät für Mathematik, Regensburg, Germany

    Felix Finster

  2. Regensburg, Germany, Fakultät für Mathematik Universität Rege, Regensburg, Germany

    Johannes Kleiner

  3. Regensburg, Germany, Fakultät für Mathematik Universität Rege, Regensburg, Germany

    Christian Röken

  4. Leipzig, Germany, MPI für Mathematik in den Naturwissensch, Leipzig, Germany

    Jürgen Tolksdorf

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Grosse, H., Wulkenhaar, R. (2016). A Solvable Four-Dimensional QFT. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds) Quantum Mathematical Physics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-26902-3_8

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